Annals of Statistics

Statistical Estimation and Optimal Recovery

David L. Donoho

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New formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random Gaussian noise. The formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class. It is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules, for a variety of loss functions. It is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated. The method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery.

Article information

Ann. Statist., Volume 22, Number 1 (1994), 238-270.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62C20: Minimax procedures
Secondary: 62G07: Density estimation 41A25: Rate of convergence, degree of approximation 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Bounded normal mean estimation of linear functionals confidence statements for linear functionals modulus of continuity minimax risk nonparametric regression density estimation


Donoho, David L. Statistical Estimation and Optimal Recovery. Ann. Statist. 22 (1994), no. 1, 238--270. doi:10.1214/aos/1176325367.

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